# Synthetic Proof Terms and Receipt Obligations, Not Lean Execution

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261019-r07-lean-proof-term-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h0","type":"P0"}], term={"kind":"var","name":"h0"}, claimed_type="P0". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P0, P0, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P0, P0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0001/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 2. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={"kind":"lam","name":"x1","var_type":"P1","body":{"kind":"var","name":"x1"}}, claimed_type=["imp","P1","P1"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([imp, P1, P1], [imp, P1, P1], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P1, P1], [imp, P1, P1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0002/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 3. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h2","type":"P2"},{"name":"q2","type":"Q2"}], term={"kind":"pair","left":{"kind":"var","name":"h2"},"right":{"kind":"var","name":"q2"}}, claimed_type=["and","P2","Q2"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([and, P2, Q2], [and, P2, Q2], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P2, Q2], [and, P2, Q2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0003/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 4. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"pair3","type":["and","P3","Q3"]}], term={"kind":"fst","pair":{"kind":"var","name":"pair3"}}, claimed_type="P3". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P3, P3, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P3, P3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0004/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 5. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h4","type":"P4"}], term={"kind":"var","name":"h4"}, claimed_type="P4". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P4, P4, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P4, P4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0005/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 6. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={"kind":"lam","name":"x5","var_type":"P5","body":{"kind":"var","name":"x5"}}, claimed_type=["imp","P5","P5"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([imp, P5, P5], [imp, P5, P5], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P5, P5], [imp, P5, P5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0006/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 7. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h6","type":"P6"},{"name":"q6","type":"Q6"}], term={"kind":"pair","left":{"kind":"var","name":"h6"},"right":{"kind":"var","name":"q6"}}, claimed_type=["and","P6","Q6"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([and, P6, Q6], [and, P6, Q6], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P6, Q6], [and, P6, Q6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0007/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 8. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"pair7","type":["and","P7","Q7"]}], term={"kind":"fst","pair":{"kind":"var","name":"pair7"}}, claimed_type="P7". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P7, P7, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P7, P7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0008/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 9. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h8","type":"P8"}], term={"kind":"var","name":"h8"}, claimed_type="P8". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P8, P8, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P8, P8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0009/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 10. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={"kind":"lam","name":"x9","var_type":"P9","body":{"kind":"var","name":"x9"}}, claimed_type=["imp","P9","P9"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([imp, P9, P9], [imp, P9, P9], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P9, P9], [imp, P9, P9], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0010/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 11. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h10","type":"P10"},{"name":"q10","type":"Q10"}], term={"kind":"pair","left":{"kind":"var","name":"h10"},"right":{"kind":"var","name":"q10"}}, claimed_type=["and","P10","Q10"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([and, P10, Q10], [and, P10, Q10], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P10, Q10], [and, P10, Q10], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0011/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 12. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"pair11","type":["and","P11","Q11"]}], term={"kind":"fst","pair":{"kind":"var","name":"pair11"}}, claimed_type="P11". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P11, P11, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P11, P11, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0012/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 13. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h12","type":"P12"}], term={"kind":"var","name":"h12"}, claimed_type="P12". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P12, P12, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P12, P12, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0013/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 14. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={"kind":"lam","name":"x13","var_type":"P13","body":{"kind":"var","name":"x13"}}, claimed_type=["imp","P13","P13"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([imp, P13, P13], [imp, P13, P13], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P13, P13], [imp, P13, P13], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0014/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 15. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h14","type":"P14"},{"name":"q14","type":"Q14"}], term={"kind":"pair","left":{"kind":"var","name":"h14"},"right":{"kind":"var","name":"q14"}}, claimed_type=["and","P14","Q14"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([and, P14, Q14], [and, P14, Q14], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P14, Q14], [and, P14, Q14], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0015/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 16. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"pair15","type":["and","P15","Q15"]}], term={"kind":"fst","pair":{"kind":"var","name":"pair15"}}, claimed_type="P15". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P15, P15, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P15, P15, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0016/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 17. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h16","type":"P16"}], term={"kind":"var","name":"h16"}, claimed_type="P16". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P16, P16, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P16, P16, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0017/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 18. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={"kind":"lam","name":"x17","var_type":"P17","body":{"kind":"var","name":"x17"}}, claimed_type=["imp","P17","P17"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([imp, P17, P17], [imp, P17, P17], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P17, P17], [imp, P17, P17], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0018/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 19. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h18","type":"P18"},{"name":"q18","type":"Q18"}], term={"kind":"pair","left":{"kind":"var","name":"h18"},"right":{"kind":"var","name":"q18"}}, claimed_type=["and","P18","Q18"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([and, P18, Q18], [and, P18, Q18], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P18, Q18], [and, P18, Q18], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0019/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 20. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"pair19","type":["and","P19","Q19"]}], term={"kind":"fst","pair":{"kind":"var","name":"pair19"}}, claimed_type="P19". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P19, P19, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P19, P19, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0020/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 21. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h20","type":"P20"}], term={"kind":"var","name":"h20"}, claimed_type="P20". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P20, P20, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P20, P20, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0021/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 22. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[], term={"kind":"lam","name":"x21","var_type":"P21","body":{"kind":"var","name":"x21"}}, claimed_type=["imp","P21","P21"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([imp, P21, P21], [imp, P21, P21], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([imp, P21, P21], [imp, P21, P21], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0022/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 23. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h22","type":"P22"},{"name":"q22","type":"Q22"}], term={"kind":"pair","left":{"kind":"var","name":"h22"},"right":{"kind":"var","name":"q22"}}, claimed_type=["and","P22","Q22"]. Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation([and, P22, Q22], [and, P22, Q22], Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([and, P22, Q22], [and, P22, Q22], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0023/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 24. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"pair23","type":["and","P23","Q23"]}], term={"kind":"fst","pair":{"kind":"var","name":"pair23"}}, claimed_type="P23". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P23, P23, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P23, P23, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0024/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 25. Synthetic Implication/Conjunction Term Type against a Claim

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h24","type":"P24"}], term={"kind":"var","name":"h24"}, claimed_type="P24". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?).

View answer and explanation(P24, P24, Yes)

Apply only the specified small-calculus rules. This synthetic type inference is not a Lean kernel receipt. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (P24, P24, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0025/)

Proof term · qf-ext19-proofterm-small-calculus · candidate





## 26. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h0","type":"P0"}], term={"kind":"var","name":"h0"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h0], [h0], [], P0, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h0], [h0], [], P0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0026/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 27. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h1","type":"P1"}], term={"kind":"var","name":"missing1"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing1], [h1], [missing1], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing1], [h1], [missing1], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0027/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 28. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h2","type":"P2"}], term={"kind":"var","name":"h2"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h2], [h2], [], P2, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h2], [h2], [], P2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0028/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 29. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h3","type":"P3"}], term={"kind":"var","name":"missing3"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing3], [h3], [missing3], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing3], [h3], [missing3], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0029/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 30. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h4","type":"P4"}], term={"kind":"var","name":"h4"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h4], [h4], [], P4, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h4], [h4], [], P4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0030/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 31. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h5","type":"P5"}], term={"kind":"var","name":"missing5"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing5], [h5], [missing5], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing5], [h5], [missing5], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0031/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 32. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h6","type":"P6"}], term={"kind":"var","name":"h6"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h6], [h6], [], P6, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h6], [h6], [], P6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0032/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 33. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h7","type":"P7"}], term={"kind":"var","name":"missing7"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing7], [h7], [missing7], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing7], [h7], [missing7], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0033/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 34. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h8","type":"P8"}], term={"kind":"var","name":"h8"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h8], [h8], [], P8, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h8], [h8], [], P8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0034/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 35. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h9","type":"P9"}], term={"kind":"var","name":"missing9"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing9], [h9], [missing9], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing9], [h9], [missing9], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0035/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 36. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h10","type":"P10"}], term={"kind":"var","name":"h10"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h10], [h10], [], P10, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h10], [h10], [], P10, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0036/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 37. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h11","type":"P11"}], term={"kind":"var","name":"missing11"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing11], [h11], [missing11], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing11], [h11], [missing11], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0037/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 38. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h12","type":"P12"}], term={"kind":"var","name":"h12"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h12], [h12], [], P12, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h12], [h12], [], P12, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0038/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 39. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h13","type":"P13"}], term={"kind":"var","name":"missing13"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing13], [h13], [missing13], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing13], [h13], [missing13], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0039/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 40. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h14","type":"P14"}], term={"kind":"var","name":"h14"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h14], [h14], [], P14, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h14], [h14], [], P14, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0040/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 41. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h15","type":"P15"}], term={"kind":"var","name":"missing15"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing15], [h15], [missing15], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing15], [h15], [missing15], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0041/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 42. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h16","type":"P16"}], term={"kind":"var","name":"h16"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h16], [h16], [], P16, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h16], [h16], [], P16, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0042/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 43. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h17","type":"P17"}], term={"kind":"var","name":"missing17"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing17], [h17], [missing17], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing17], [h17], [missing17], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0043/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 44. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h18","type":"P18"}], term={"kind":"var","name":"h18"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h18], [h18], [], P18, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h18], [h18], [], P18, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0044/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 45. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h19","type":"P19"}], term={"kind":"var","name":"missing19"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing19], [h19], [missing19], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing19], [h19], [missing19], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0045/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 46. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h20","type":"P20"}], term={"kind":"var","name":"h20"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h20], [h20], [], P20, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h20], [h20], [], P20, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0046/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 47. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h21","type":"P21"}], term={"kind":"var","name":"missing21"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing21], [h21], [missing21], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing21], [h21], [missing21], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0047/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 48. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h22","type":"P22"}], term={"kind":"var","name":"h22"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h22], [h22], [], P22, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h22], [h22], [], P22, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0048/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 49. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h23","type":"P23"}], term={"kind":"var","name":"missing23"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([missing23], [h23], [missing23], not-evaluated, No)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([missing23], [h23], [missing23], not-evaluated, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0049/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 50. Synthetic Free Variables before Type Inference

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"h24","type":"P24"}], term={"kind":"var","name":"h24"}. Give (sorted distinct free variables, sorted context names, missing free variables, inferred type or not-evaluated, context complete?). Use the specified English-locale name order.

View answer and explanation([h24], [h24], [], P24, Yes)

Do not infer a term’s type until every free variable is bound; closed lambdas bind their own variable. This remains a synthetic calculation, not actual Lean. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([h24], [h24], [], P24, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0050/)

Proof term · qf-ext19-proofterm-context-use · candidate





## 51. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=false, term_present=false, unresolved_goals=0. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(No, No, No, 0, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, No, No, 0, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0051/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 52. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=1. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 1, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 1, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0052/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 53. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=2. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 2, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 2, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0053/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 54. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=3. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, No, 3, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 3, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0054/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 55. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=true, unresolved_goals=4. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, No, Yes, 4, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, Yes, 4, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0055/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 56. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=true, term_present=true, unresolved_goals=5. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(No, Yes, Yes, 5, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, Yes, Yes, 5, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0056/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 57. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=6. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, No, 6, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 6, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0057/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 58. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=7. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 7, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 7, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0058/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 59. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=true, unresolved_goals=8. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, No, Yes, 8, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, Yes, 8, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0059/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 60. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=9. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, No, 9, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 9, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0060/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 61. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=true, term_present=true, unresolved_goals=10. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(No, Yes, Yes, 10, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, Yes, Yes, 10, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0061/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 62. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=11. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 11, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 11, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0062/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 63. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=false, unresolved_goals=12. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, No, No, 12, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, No, 12, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0063/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 64. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=13. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 13, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 13, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0064/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 65. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=14. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 14, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 14, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0065/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 66. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=true, term_present=false, unresolved_goals=15. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(No, Yes, No, 15, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, Yes, No, 15, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0066/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 67. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=true, unresolved_goals=16. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, No, Yes, 16, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, Yes, 16, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0067/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 68. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=17. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 17, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 17, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0068/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 69. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=18. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, No, 18, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 18, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0069/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 70. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=19. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 19, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 19, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0070/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 71. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=false, tactic_finished=false, term_present=true, unresolved_goals=20. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(No, No, Yes, 20, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (No, No, Yes, 20, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0071/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 72. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=false, unresolved_goals=21. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, No, 21, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, No, 21, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0072/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 73. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=22. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 22, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 22, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0073/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 74. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=true, term_present=true, unresolved_goals=23. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, Yes, Yes, 23, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, Yes, Yes, 23, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0074/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 75. Synthetic Tactic Completion Is Not a Kernel Receipt

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. statement_ready=true, tactic_finished=false, term_present=false, unresolved_goals=24. Give (those three flags, goal count, tactic finished with zero goals?, term candidate?, kernel checked?, no-kernel receipt boundary). The kernel flag remains false in this protocol.

View answer and explanation(Yes, No, No, 24, No, No, No, no-kernel receipt)

A tactic candidate needs a ready statement, a finished zero-goal tactic and a term. These supplied flags are not actual Lean execution. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (Yes, No, No, 24, No, No, No, no-kernel receipt).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0075/)

Proof term · qf-ext19-proofterm-tactic-vs-term · candidate





## 76. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"","statement_id":"","term_sha256":"bad","type_sha256":"bad","kernel_checked":true,"axioms_recorded":false}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(missing, missing, No, No, Yes, No, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, missing, No, No, Yes, No, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0076/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 77. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-1","statement_id":"thm-1","term_sha256":"0000000000000000000000000000000000000000000000000000000000000001","type_sha256":"0000000000000000000000000000000000000000000000000000000000000065","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-1, thm-1, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-1, thm-1, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0077/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 78. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-2","statement_id":"thm-2","term_sha256":"0000000000000000000000000000000000000000000000000000000000000002","type_sha256":"0000000000000000000000000000000000000000000000000000000000000066","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-2, thm-2, Yes, Yes, Yes, Yes, Yes, synthetic-receipt-fields-complete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-2, thm-2, Yes, Yes, Yes, Yes, Yes, synthetic-receipt-fields-complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0078/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 79. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-3","statement_id":"thm-3","term_sha256":"0000000000000000000000000000000000000000000000000000000000000003","type_sha256":"bad","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-3, thm-3, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-3, thm-3, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0079/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 80. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-4","statement_id":"thm-4","term_sha256":"bad","type_sha256":"0000000000000000000000000000000000000000000000000000000000000068","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-4, thm-4, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-4, thm-4, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0080/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 81. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-5","statement_id":"","term_sha256":"0000000000000000000000000000000000000000000000000000000000000005","type_sha256":"0000000000000000000000000000000000000000000000000000000000000069","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-5, missing, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-5, missing, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0081/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 82. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"","statement_id":"thm-6","term_sha256":"0000000000000000000000000000000000000000000000000000000000000006","type_sha256":"bad","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(missing, thm-6, Yes, No, Yes, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, thm-6, Yes, No, Yes, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0082/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 83. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-7","statement_id":"thm-7","term_sha256":"0000000000000000000000000000000000000000000000000000000000000007","type_sha256":"000000000000000000000000000000000000000000000000000000000000006b","kernel_checked":false,"axioms_recorded":false}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-7, thm-7, Yes, Yes, No, No, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-7, thm-7, Yes, Yes, No, No, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0083/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 84. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-8","statement_id":"thm-8","term_sha256":"bad","type_sha256":"000000000000000000000000000000000000000000000000000000000000006c","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-8, thm-8, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-8, thm-8, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0084/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 85. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-9","statement_id":"thm-9","term_sha256":"0000000000000000000000000000000000000000000000000000000000000009","type_sha256":"bad","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-9, thm-9, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-9, thm-9, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0085/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 86. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-10","statement_id":"","term_sha256":"000000000000000000000000000000000000000000000000000000000000000a","type_sha256":"000000000000000000000000000000000000000000000000000000000000006e","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-10, missing, Yes, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-10, missing, Yes, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0086/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 87. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-11","statement_id":"thm-11","term_sha256":"000000000000000000000000000000000000000000000000000000000000000b","type_sha256":"000000000000000000000000000000000000000000000000000000000000006f","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-11, thm-11, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-11, thm-11, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0087/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 88. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"","statement_id":"thm-12","term_sha256":"bad","type_sha256":"bad","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(missing, thm-12, No, No, Yes, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, thm-12, No, No, Yes, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0088/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 89. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-13","statement_id":"thm-13","term_sha256":"000000000000000000000000000000000000000000000000000000000000000d","type_sha256":"0000000000000000000000000000000000000000000000000000000000000071","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-13, thm-13, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-13, thm-13, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0089/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 90. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-14","statement_id":"thm-14","term_sha256":"000000000000000000000000000000000000000000000000000000000000000e","type_sha256":"0000000000000000000000000000000000000000000000000000000000000072","kernel_checked":true,"axioms_recorded":false}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-14, thm-14, Yes, Yes, Yes, No, Yes, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-14, thm-14, Yes, Yes, Yes, No, Yes, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0090/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 91. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-15","statement_id":"","term_sha256":"000000000000000000000000000000000000000000000000000000000000000f","type_sha256":"bad","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-15, missing, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-15, missing, Yes, No, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0091/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 92. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-16","statement_id":"thm-16","term_sha256":"bad","type_sha256":"0000000000000000000000000000000000000000000000000000000000000074","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-16, thm-16, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-16, thm-16, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0092/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 93. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-17","statement_id":"thm-17","term_sha256":"0000000000000000000000000000000000000000000000000000000000000011","type_sha256":"0000000000000000000000000000000000000000000000000000000000000075","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-17, thm-17, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-17, thm-17, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0093/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 94. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"","statement_id":"thm-18","term_sha256":"0000000000000000000000000000000000000000000000000000000000000012","type_sha256":"bad","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(missing, thm-18, Yes, No, Yes, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, thm-18, Yes, No, Yes, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0094/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 95. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-19","statement_id":"thm-19","term_sha256":"0000000000000000000000000000000000000000000000000000000000000013","type_sha256":"0000000000000000000000000000000000000000000000000000000000000077","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-19, thm-19, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-19, thm-19, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0095/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 96. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-20","statement_id":"","term_sha256":"bad","type_sha256":"0000000000000000000000000000000000000000000000000000000000000078","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-20, missing, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-20, missing, No, Yes, Yes, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0096/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 97. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-21","statement_id":"thm-21","term_sha256":"0000000000000000000000000000000000000000000000000000000000000015","type_sha256":"bad","kernel_checked":false,"axioms_recorded":false}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-21, thm-21, Yes, No, No, No, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-21, thm-21, Yes, No, No, No, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0097/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 98. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-22","statement_id":"thm-22","term_sha256":"0000000000000000000000000000000000000000000000000000000000000016","type_sha256":"000000000000000000000000000000000000000000000000000000000000007a","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-22, thm-22, Yes, Yes, Yes, Yes, Yes, synthetic-receipt-fields-complete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-22, thm-22, Yes, Yes, Yes, Yes, Yes, synthetic-receipt-fields-complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0098/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 99. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"env-23","statement_id":"thm-23","term_sha256":"0000000000000000000000000000000000000000000000000000000000000017","type_sha256":"000000000000000000000000000000000000000000000000000000000000007b","kernel_checked":false,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(env-23, thm-23, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (env-23, thm-23, Yes, Yes, No, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0099/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate





## 100. Synthetic Kernel Metadata Qualification and Axiom Inventory Flag

Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Supplied metadata={"environment_id":"","statement_id":"thm-24","term_sha256":"bad","type_sha256":"bad","kernel_checked":true,"axioms_recorded":true}. Give (environment or missing, statement or missing, term-hash lexical shape?, type-hash lexical shape?, kernel flag, axioms-recorded flag, nonempty IDs plus both shapes plus kernel flag qualify?, synthetic receipt fields complete/still incomplete). Completeness additionally requires the axiom-recorded flag; no digest or kernel run is independently verified here.

View answer and explanation(missing, thm-24, No, No, Yes, Yes, No, synthetic-receipt-still-incomplete)

Keep qualification and recorded axioms separate. Synthetic metadata completeness is not an actual Lean kernel acceptance or semantic-trust certificate. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (missing, thm-24, No, No, Yes, Yes, No, synthetic-receipt-still-incomplete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r07-lean-proof-term-0100/)

Proof term · qf-ext19-proofterm-kernel-receipt · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261019-r07-lean-proof-term-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261019-r07-lean-proof-term-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261019-r07-lean-proof-term-foundations/validation-report.json)